G10 Write a line equation from a graph, point, or slope

Ontario Grade 10 Mathematics

Modelling Linear Relations

Use a graph, a point and slope, or a known slope and intercept

A line equation describes every point on a straight line. You may be given the line on a graph, one point and its slope, or information about the line's slope and intercept. In each case, the goal is to find an equation that matches the information. This lesson reviews how to read coordinates and slope, then connects those ideas to a line equation.

What you will learn

Recognize slope and the vertical intercept in a line equation.
Find slope from two points on a graph.
Write a line equation using a graph, or a point and slope.
Decide when the available information is not enough to identify one line.

1. Grade 9 bridge: coordinates and slope

A coordinate is an ordered pair that names a point on a graph. In

(x,y)

, the first number tells how far to move horizontally, and the second tells how far to move vertically. For example,

(3,2)

is 3 units right and 2 units up from the origin.

Slope describes how steep a line is and whether it rises or falls as you move from left to right. You can find it by comparing the vertical change with the horizontal change between two points. Vertical change is also called rise; horizontal change is also called run. Keep the changes in the same order for both coordinates.

A positive slope means the line rises from left to right. A negative slope means it falls. A horizontal line has slope 0. A vertical line has no defined slope because its horizontal change is 0, and division by 0 is not allowed.

m=\frac{y_2-y_1}{x_2-x_1}

Read coordinates in the order horizontal, then vertical.
Slope compares vertical change to horizontal change.
A graph's scale matters: count the labelled units, not just the grid spaces.

2. The line equation and what its parts mean

A useful form for a line equation is

y=mx+b

. Here,

x

and

y

are coordinates of points on the line. The number

m

is the slope. The number

b

is the

y

-intercept: the

y

-value where the line crosses the vertical axis. At that crossing,

x=0

.

For example, if a line crosses the vertical axis at 1 and rises 2 units for every 1 unit right, its slope is 2 and its intercept is 1. The equation is

y=2x+1

. A point such as

(1,3)

fits because replacing

x

with 1 gives

y=3

.

A graph gives a line's equation when you can identify its slope and its

y

-intercept. If the intercept is hard to read, choose two clear points on the line to find the slope. Then use a known point to find the intercept. A point is on the line only if it lies exactly on the drawn line, not merely near it.

A slope by itself does not usually identify one unique line. Many lines can have the same slope but cross the vertical axis at different places. To write one specific equation, you also need an intercept or a point on the line.

y=mx+b

In $y=mx+b$ , $m$ is slope and $b$ is the vertical-axis intercept.
A graph can provide the slope and intercept directly or through points on the line.
A slope and a point are enough to identify a line; slope alone is not.

3. From a graph, point, or slope to an equation

From a graph, first read the

y

-intercept if it is clear. Next, find slope using two points on the line. You may choose the intercept as one of the points. Count the vertical and horizontal changes, and keep their signs: a move down is negative, and a move left is negative. Then place the slope and intercept in

y=mx+b

.

If you know a point and the slope, use the same equation. Replace

x

and

y

with the point's coordinates and replace

m

with the given slope. Solve for

b

. Then write the complete equation using the values of

m

and

b

.

If you are given slope and

y

-intercept, substitute them directly into

y=mx+b

. If only the slope is given, explain that more information is needed. There is no single answer unless a point or intercept is also known.

Check your equation by putting a known point into it. Both sides should agree. Also compare the sign and steepness of the slope with the graph. This check can catch a reversed rise and run or a sign error.

b=y-mx

For a graph: find the intercept and slope, then substitute.
For a point and slope: substitute the point to find the intercept.
Check that a supplied point satisfies the final equation.

4. Guided example and independent practice

The table shows a few points on a line. It is a simple way to organize the coordinates read from a graph. The change in

y

is 2 while the change in

x

is 1, so the slope is 2. The point at

x=0

gives the intercept. The worked example uses both facts to write and check the equation.

For independent practice, try these without looking at the quick check first. (1) A line has slope 3 and passes through

(2,7)

. Write its equation. (2) A line crosses the vertical axis at

-2

and has slope

\frac{1}{2}

. Write its equation. (3) A line has slope

-1

and crosses the vertical axis at 4. Explain how you can check a point on your graph against its equation.

For each practice problem, identify what information is given before calculating. If a point and slope are given, find the intercept. If the intercept and slope are given, substitute them directly. In every case, use the equation to test a known point.

A table of points can make the graph's changes easier to compare.
When a point and slope are given, calculate the intercept before writing the final equation.
Use substitution to check the equation against a point.

Points on the line in the worked example

x	y
0	1
1	3
2	5

Worked example

Read a line from its points

A line passes through the points in the table. Write an equation for the line.

Find the slope
From $(0,1)$ to $(2,5)$ , the horizontal change is 2 and the vertical change is 4. Divide the vertical change by the horizontal change.
$m=\frac{5-1}{2-0}=2$
Read the intercept
The point $(0,1)$ is where the line has $x=0$ . Its $y$ -value is therefore the $y$ -intercept, so $b=1$ .
$b=1$
Write the equation
Substitute the slope and intercept into $y=mx+b$ . This gives an equation for every point on the line.
$y=2x+1$
Check a second point
Use $x=2$ in the equation. The result is $y=5$ , which matches the point $(2,5)$ from the table.
$2(2)+1=5$

Answer: The line's equation is

y=2x+1

.

Check: Both listed points fit: when

x=0

,

y=1

; when

x=2

,

y=5

.

Common mistakes and how to avoid them

Reversing rise and run when finding slope.

Correction: Use vertical change divided by horizontal change, and keep the same point order in both differences.

Reading the first coordinate as the vertical position.

Correction: In

(x,y)

,

x

is horizontal and

y

is vertical.

Using the slope as the intercept.

Correction: In

y=mx+b

, slope is

m

and the vertical-axis intercept is

b

.

Writing one equation when only the slope is known.

Correction: Ask for a point on the line or its intercept. Different intercepts give different lines with the same slope.

Lesson summary

Slope is vertical change divided by horizontal change.
In $y=mx+b$ , $m$ gives slope and $b$ gives the $y$ -intercept.
A graph can provide points, slope, and intercept. Use the information it shows.
A point and slope let you find the intercept and write the equation.
Check a proposed equation by substituting a point on the line.

Check your understanding

Question 1

A line has slope 2 and

y

-intercept

-3

. Which is its equation?

$y=2x-3$
$y=-3x+2$
$y=2x+3$
$y=-2x-3$

Show answer and explanation

y=2x-3

In

y=mx+b

, substitute

m=2

and

b=-3

.

Question 2

A line has slope 3 and passes through

(1,5)

. Which equation fits?

$y=3x+2$
$y=3x+5$
$y=2x+3$
$y=-3x+8$

Show answer and explanation

y=3x+2

Substituting the point gives

5=3(1)+b

, so

b=2

.

Question 3

A line passes through

(0,4)

and

(2,0)

. What is its slope?

$-2$
$2$
$\frac{1}{2}$
$-\frac{1}{2}$

Show answer and explanation

-2

The vertical change is

0-4=-4

and the horizontal change is

2-0=2

. Their quotient is

-2

.

Question 4

You know only that a line has slope

-1

. What else would identify one specific line?

A point on the line or its $y$ -intercept
The slope written as a fraction
The direction of the $x$ -axis
The name of the line

Show answer and explanation

A point on the line or its

y

-intercept

Many lines can share slope

-1

. A point or intercept tells which one is intended.

Key terms

Coordinate: A pair of numbers, written $(x,y)$ , that names a point on a graph.
Slope: A measure of a line's steepness, found by dividing vertical change by horizontal change.
Rise: The vertical change between two points.
Run: The horizontal change between two points.
$y$ -intercept: The point where a line crosses the vertical axis; its $x$ -coordinate is 0.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic G10. It is a study resource, not an official curriculum publication.

Official curriculum reference

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G10 · Write a line equation from a graph, point, or slope

Ontario Grade 10 Mathematics

Use a graph, a point and slope, or a known slope and intercept

What you will learn

1. Grade 9 bridge: coordinates and slope

2. The line equation and what its parts mean

3. From a graph, point, or slope to an equation

4. Guided example and independent practice

Points on the line in the worked example

Read a line from its points

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MFM2P

About this lesson